Mode decomposition of three-dimensional mixedmode cracks via two-state integrals

Transcription

1 Mode decomposition of three-dimensional mixedmode cracs via two-state integrals Young ong im*, Hyun-Gyu im** Seyoung m*** Department of Mechanical Engineering orea Advanced nstitute of Science Technology (AST) Science Town, Taejon, South orea * Graduate Student (yjim@imhp.aist.ac.r) ** Post-doctoral researcher (hg@imhp.aist.ac.r) *** Professor (sim@mail.aist.ac.r) (All correspondence should be addressed to Seyoung m) Tel. : Fax. : Abstract A numerical scheme is proposed to obtain the individual stress intensity factors in an axisymmetric crac in a three-dimensional mixed-mode crac. The procedures presented here are based on the path independence of M integrals mutual or two-state conservation integrals, which involve two elastic fields. A useful method to decompose the stress intensity factors along curved three-dimensional cracs under mixed mode is derived by using appropriate auxiliary fields for the plane problems. The choice of the auxiliary fields available is critical to success of the present scheme, in this study it is made of not only the asymptotic plane-strain solution, which requires some remedy in application of the twostate integral due to the lac of equilibrium compatibility, but a numerical solution with a given stress intensity as well. Some numerical examples of penny shaped cracs are presented to investigate the applicability effectiveness of the method for problems of axisymmetric three-dimensional cracs. eywords: Mode decomposition; Stress intensity factors; integral; Mutual integral; Auxiliary field; Three-dimensional crac; Penny shaped crac

2 . ntroduction Conservation integrals in elasticity have been widely applied to the fracture mechanics, among which the integral is the most popular one. The integral is path-independent has been shown to be identical to rwin s energy release rate associated with the collinear extension of a crac in an elastic solid (Rice, 968). t has been related to the crac tip stress intensity factors in both linear nonlinear elastic solids subjected to infinitesimal deformations (Hutchinson, 968). t is necessary to evaluate the individual stress intensity factors separately for mixed-mode crac problems in order to investigate the crac growth the crac propagation. However, the evaluation of the integral alone does not determine the individual stress intensity factors,, separately. Many wors on mixed-mode crac problems using the path-independent integrals have been reported. Bui (983) developed a technique using the integral associated with Mode Mode, in which the symmetric antisymmetric parts of the planar displacement, strain stress fields about the crac plane are separated. Stern et al. (976) employed another conservation integral based on Betti s reciprocal wor theorem with nown auxiliary fields. Later the method was extended for the straight interfacial cracs by Hong Stern (978). For planar cracs, Yau et al. (980), Wang Yau (98) Shih Asaro (988) utilized a new class of conservation integral nown as mutual integral or two-state conservation integral, proposed by Eshelby (956) later by Chen Shield (977): Naamura Par (989) Naamura (99) employed the same approach to determine the mixed-mode stress intensity factors for three-dimensional interface crac problems. Niishov Atluri (987a) developed a domain integral approach to calculate mixed-mode stress intensity factors for planar three-dimensional cracs. Nahta Moran (993) Gosz et al. (998) used asymptotic auxiliary fields for the plane problems to decompose the stress intensity factors in mixed-mode cracs, proposed a method to evaluate the divergence term in the two-state integral. Choi Earmme (99) employed the two-state L integral to evaluate the stress intensity factors in circular arc-shaped interfacial crac. Recently m im (000) showed that the two-state M integral is applicable for computing the intensity of the singular near-tip field for a generic isotropic composite wedge including planar cracs. The main interest of fracture mechanics is shifting from two- to three-dimensional crac

3 problems particularly it becomes increasingly important to study three-dimensional mixed-mode crac problems for the crac growth propagation prediction. This study presents a method to obtain the individual stress intensity factors for axisymmetric three-dimensional cracs in mixed mode. The method is based on the path independence of M two-state integral, which involves two independent elastic fields. The path-independence of these conservation integrals enables one to obtain each stress intensity factor from the displacements stresses remote from the crac tip. n this paper, we present a simple method to evaluate the two-state integral with asymptotic auxiliary fields for the plane problems. n particular, the divergence term in the two-state integral, which arises due to the lac of equilibrium compatibility of asymptotic auxiliary fields, is reduced to a simple form, with the aid of the equilibrium equation for the plane problems, by imposing displacements of asymptotic solutions on the nodes of finite elements. Numerical auxiliary fields, finite element solutions for penny-shaped crac problems under pure Mode Mode, are used to obtain reference stress intensity factors in threedimensional mixed-mode cracs. The purpose of this study is to show the validity of the numerical auxiliary fields the effectiveness of the present method to calculate the divergence term in the two-state integral for mode decomposition utilizing the auxiliary field of the plane-strain asymptotic solution. n the next section the basic formulation for the method is described the solution procedure is established. The implementation of the scheme is explained in Section 3, numerical examples are carried out in Section 4.. Formulation of the problems.. Axisymmetric formulation n three-dimensional infinitesimal deformations of homogeneous isotropic bodies, M integral is defined as (nowles Sternberg, 97) = Wxi ni Tiui, j x j Tiu da, ( i =,,3) () M A i where W is the strain energy density, u i is the displacement, n i is the unit outward normal vector, T i is the traction vector, A indicates a surface.

4 Consider crac problems in an elastic, homogeneous, isotropic axisymmetric solid under axisymmetric loading conditions. As a result of its axisymmetry, Eq. () can then be rewritten as (uo, 987) M π Wxα nα Tα uα, β xβ Tα u xds () C = α where the subscripts α β indicate the components in the r z or the x x plane (see Fig. ), C encloses the crac tip (it may be taen to be C or C in Fig. ). Note that x x are used for r z whenever it is convenient to do so. Physically, Eq. () implies that M integral is a driving force for a crac to exp uniformly (Budiansy Rice, 973). Straightforward argument reveals that the M integral has the following relationships for the axisymmetric cracs (uo, 987) in the absence of Mode or torsion. = + (3) with M = = α π rc = M π rc = α where α ( ν )/ E = (E : Young s modulus; ν : Poisson s ratio), r c is the r or the x coordinate of the crac tip, that is, the shortest distance from the z axis to the crac tip. t should be noted here that the integral in Eq. (3) alone does not provide adequate information for determining the individual stress intensity factors in a mixedmode crac problem. Following Chen Shield (977), we consider two independent elastic states of a penny shaped crac: the field of the target problem denoted by superscript () an auxiliary field denoted by (). Let the elastic state from the superposition of the two elastic fields be denoted by superscript (0). Then the integral for the resulting state has the following form: + ( 0 ) ( ) ( ) (, ) = + (4) in which ( ), ( ) (, ) are given as 3

5 ( a) ( a) =, ( =, ) Fm nm x ds a r C c (5a) (,) (,) = Fm nm xds r (5b) C c with F ( a) m = W ( a) x δ i im σ ( a) i u ( a) i, j x δ j m σ ( a) n u ( a) δ nm F (, ) m = W (, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( σ i ui, j + σ i ui, j ) x jδ m ( σ n u σ n u ) δ nm xiδ im + where δ ij is the ronecer delta (, ) W is the two-state strain energy density of the elastic body, defined by (, ) ( ) ( ) W Cijlui, j u,l = = C u u (6) ijl ( ) i, j ( ),l n Eqs. (5a) (5b), σ ( ) ij u can be obtained from analytic or numerical solution ( ) i to the target problem along a properly selected integration path C, while σ ( ) ij u ( ) i are given from the auxiliary field, which is chosen in a convenient manner. Recalling the relationships of Eq. (3), one finds that the integral for the state (0) is expressed as () () () () {[ ] [ ] } + + (0) + which leads to = α (7) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( ) = α. (8) Comparison between Eqs. (4) (8) reveals that () () () () ( ) (,) + = α (9) The integral shown in Eqs. (5b) (9) deals with the interaction term only is to 4

6 be used for solving mixed-mode penny shaped crac problems in a linear elastic solid. t should be noted here that the deformation at the crac tip (i.e., (, ) integral is related to the details of the stresses in Eq. (9)). Due to the path independence of this integral, however, it may be evaluated in the region away from the crac tip (i.e., the integral in Eq. (5b)), where such a calculation can be carried out with greater accuracy convenience than near the crac tip. Eq. (5b) together with Eq. (9) provides, in fact, sufficient information for determining the stress intensity factors for a mixed-mode fracture problem of a penny shaped crac, when a proper nown auxiliary field is introduced. Let the superscript (a) indicate the solution for an auxiliary elastic field, wherein the body under consideration is in the state of Mode deformation only, i.e., a ) ( 0 ( a ) 0 =. (0) Eq. (9) can be simplified as (, a) () α =. () (a) For the target problem field for the auxiliary field of Mode, we have, respectively: () () {[ ] [ ] } ( a ) ( a ) = α, α[ ] () + =. () Eqs. () () leads to the expressions of the individual stress intensity factors for the target field in terms of the conservation integrals, ( ), ( a ) (,a ) : () (, a) [ ] (, a) = () () (a) = ± (a) α α 4. (3) t should be noted that the integrals ( ), ( a ) (,a ) have to be calculated accurately for proper evaluation of ( ) ( ). For a given crac geometry loading condition, this can be achieved by integrating Eqs. (5a) (5b) along a properly selected b in the far field utilizing the domain integral expression (Li et al., 985; Niishov Atluli, 987b)... Three-dimensional formulation 5

7 n three-dimensional infinitesimal deformation of homogeneous isotropic elastic bodies the energetic force integral is given as (Eshelby, 956; nowles Sternberg, 97) A ( Wn Tiui ) da H jn jda =, = (4) A where A indicates a surface surrounding the crac front as in Fig., it includes the crac surfaces as well as the remaining boundary so that A = S0 ; the integr H j is the energy momentum tensor, given as Wδ σ u. Note that the pointwise value of along j ij i, the crac front is required for mode decomposition. Let ( s ) indicate the energy release rate associated with the self-similar crac growth on the crac front. As the cylindrical surface A shrins to the crac front line (Moran Shih, 987), ( s ) is given as lim l S H jm jda Γ 0 t ( s) = (5) l ν ds Lc where Γ is the circular contour on the shrining surface r z plane that represents the cross section of the S t (see Fig. c); s denotes the coordinate along the crac front of a three-dimensional crac; l, which is short for l ( s ), indicates the component of crac advance vector (see Fig. b); ν, which is short for ν s ), represents an outward unit vector ( normal to crac front on the crac plane m j is the normal to Γ pointing towards the crac front; i.e., m j = n j on Γ as shown in Fig. a. Furthermore, c L is a small interval on the line of the crac front for virtual crac extension. As the crac tip is approached in a three-dimensional crac, asymptotically the plane-strain or the plane-stress state prevails (Rice, 968) therefore we can use the following relationships for the three-dimensional mixed-mode crac: = + + (6) with = α, = α, α =. ν 6

8 As in the axisymmetric case, the integral in Eqs. (5) (6) alone does not provide adequate information for determining the individual stress intensity factors, in a mixed-mode crac problem. Other information may be obtained from the two-state integral (Chen Shield, 977). Consider two independent elastic states of a penny shaped crac in an elastic medium, each denoted by superscript () (). Let the equilibrium state from the superposition of the two states be denoted by superscript (0). Then the integral for the superimposed state is obtained in the following form: + ( 0 ) ( ) ( ) (, ) = + (7) ( ) in which ( =,) is defined by Eq. (5), (, ) is given as (,) lim = Γ 0 St l H Lc (,) j l ν ds m dγ j (8) with H ( ) ( ) ( ) ( σ u σ u ) ) (, ) (, ) ( j = W δ j ij i, + ij i,. From Eq. (6), one finds that the integral for the elastic state (0) may be written as (0) = () + () + () () () () () () α + + (9) ν comparison to Eq. (7) yields (,) = () () () () () () α + +. (0) ν Eq. (8) together with Eq. (0) provides, in fact, sufficient information for determining the individual stress intensity factors for a mixed-mode crac when two nown auxiliary solutions are introduced. Let two auxiliary solutions be denoted by a superscript (a) (b). The auxiliary state (a) is chosen to be a pure Mode state, (b) to be a pure Mode. Then ( a ) ( b ) are written as ( a ) ( ) = α[ ] a ( b ) ( b ), [ ] = α. () ν 7

9 Together with Eqs. (6) (0), these lead to the following expression for ( ), ( ) ( ) in terms of ( ), (,a ), ( a ), (,b ), ( b ) () (, a) = (a) α (,a ) [ ] ( ) = () ( a ) 4 ( ) (, b ) E = + ν ( b ) (,b ) [ ] ( b ) ( ) = (3) 4 () = ± α () (, a) (,b ) [ ] [ ] (a) 4 4 (b) ( ) ( ) α[ ] =. (4) Note that the path-independent integrals on the right h side of the above expressions may be calculated accurately, transforming Eq. (8) into the domain integral expression. The values of ( s ) the two-state integral along a three-dimensional crac front are given by the limiting contour integral as seen in Eqs. (5) (8). Following Moran Shih (987), we can show that the domain integral representation, which is more suitable for numerical computation, is obtained as ( s) = ( H q H q ) + V L c j, j j, j l ( s) ν ( s) ds dv (5) where q is the weight function, V is the domain bounded by S t S 0 (see Fig. c), L c is the crac front line from s a to s b (see Fig. b). t should be noted that S t must shrin onto the crac tip in order to evaluate the pointwise value of along the crac front. The weight function is defined as follows q is smooth enough for the indicated operations to be carried out l on St q = 0 on S 0 (6) otherwise arbitrary. The divergence term H, in Eq. (5) vanishes if the auxiliary fields satisfy the equilibrium j j 8

10 compatibility in the absence of body force, thermal stress, inertia. The success of the present scheme crucially relies upon the availability of the auxiliary solutions. For two-dimensional crac problems, we see that the three modes of the -fields, which have inverse square root singularity, are valid over the entire domain, therefore they naturally come up as a set of three independent auxiliary solutions. However, there exist no simple auxiliary solutions available for three-dimensional cracs with a curved line of crac front or tip. n spite of such a limitation, however, the present scheme is applicable for three-dimensional cracs once we have any auxiliary solution valid only on the local region where the domain integral is carried out. For such auxiliary solutions, we may therefore choose the solutions for an axisymmetric penny shaped crac with the same radius as the radius of curvature of the local crac front line in the target problem of a three-dimensional crac for which we want to decompose the fracture modes. The solution for a penny shaped crac in an infinite body is given in the integral form (Hartranft Sih, 973; assir Sih, 974) so not convenient for numerical computation. n order to obtain auxiliary fields, we tae finite element solutions to penny shaped crac problems under pure Mode Mode. We demonstrate in the next section the validity of the numerical auxiliary fields through some numerical examples. Another choice of auxiliary field is the use of the plane-strain asymptotic solution. Away from a curved crac front the plane-strain asymptotic solution fails to meet the equilibrium the compatibility, so arises non-zero divergence term H, in Eq. (5) for a curved three-dimensional crac. n j j general, this divergence term should be included in the calculation of the two-state integral to decompose the mixed modes in a curved three-dimensional crac, because the auxiliary fields in the form of the asymptotic plane-strain solution or -field do not satisfy the equilibrium compatibility in three-dimensional domain with curved cracs. n other words, the threedimensional effect should be considered with the divergence term retained in the two-state integral. n particular, it is crucial to incorporate the divergence term in the two-state integral for a highly curved crac. The divergence term in the two-state integral can be written as H = σ ε σ u σ u (7) (,) () () () () () () j, j ij ij, ij i, j ij, j i, in which superscript () indicates an auxiliary field. The higher order gradients in Eq. (7) give rise to the difficulty in evaluating the divergence term. Nahta Moran (993) Gosz et al. (998) presented a method to evaluate the divergence term by introducing 9

11 curvilinear coordinates in the expression of deformation gradients. n this paper, we propose a simple method to impose the plane-strain asymptotic solutions, as auxiliary fields, on threedimensional domain. The displacements of two-dimensional asymptotic solution are imposed on the nodes of three-dimensional finite element model, then the first second terms in Eq. (7) cancel each other because displacement field satisfies compatibility condition; i.e., σ ε = σ u. n the present method, the stresses of the asymptotic solutions are used in () ij () ij, () ij () i, j the expression of the two-state integral, the strains are evaluated at the integration points by using finite element interpolation from the nodal values of the two-dimensional asymptotic solution. The equilibrium equation in the absent of body force in the cylindrical coordinates shown in Fig. 3 can be reduced to the following forms: r -direction : σ r () rr () () () () () σ σ rθ rz σ rr σ θθ σ rr σ = ρ θ z ρ ρ () θθ (8a) θ -direction : σ r r () θ () σ θθ + ρ θ σ + z () () σ rθ σ θ + = ρ ρ () θz r (8b) z -direction : σ r () rz σ θz + ρ θ σ + z () σ rz + ρ () σ = ρ () () zz rz (8c) where ρ is the radius on the plane parallel to the crac surface as shown in Fig. 3. n this figure, the line p p is perpendicular to the plane of crac surface, the cylindrical coordinates are defined at the point c lying on the crac front. The expressions inside bracet in Eqs. (8a), (8b) (8c) indicate the equilibrium for the asymptotic solution in two-dimensional domain. Consequently, the divergence term in the present method can be written as H = σ u (,) () () j, j ij, j i, = () () () [ u u u ],, 3, e e e 3 e r e e r r e e e 3 e θ e e θ θ e e e 3 e e e z z z () () ( σ σ ) rr σ σ θθ () rθ () rz / ρ / ρ / ρ (9) 0

12 where the basis vectors ( e, e e ) ( e, eθ, ) in (, x x ), 3 r e z x ( r, θ, z), coordinates, respectively, are introduced for coordinate transformation. For example, we have the following expression for the divergence term in the two-state integral when the crac surface lies in the x x plane: 3 H σ () () () () () () (,) rr θθ rθ () rr θθ rθ j, j = cosθ sinθ u, sinθ + cos σ ρ () rz σ ρ u () 3, σ ρ σ σ ρ σ ρ θ u (), (30) where θ is the angle between x -axis r -axis in the x x plane. t should be emphasized that the expression (9) or (30) is easily calculated from the stresses ( σ σ, σ σ ) rr, of the plane-strain asymptotic field. θθ rθ, rz 3. Finite element implementation 3.. Axisymmetric finite element implementation n the foregoing we have established the mode decomposition method for axisymmetric three-dimensional cracs under mixed mode loading. Accuracy of the method described in Section depends on how accurately the conservation integrals are evaluated. The integration path may be chosen along the element boundaries or through the Gaussian points. Li et al. (985), Niishov Atluli (987b), Shih Asaro (988) have introduced appropriate weighting functions to obtain area/domain representation of the integral. They concluded that a very accurate value of is obtained using the domain representation the value of so obtained is insensitive to the types of weighting function. n an analogous manner it is possible to recast the line integral of Eq. (5) into the area integral: where q = F mx da (3) r A x c m A is an area enclosed by C C as shown in Fig. the function F m is given in Eqs. (5a,b). Moreover, q is a weight function which has the value of zero on the

13 contour C one on the inner contour C. Use is made of the equilibrium equation divergence theorem in deriving the expression (3). 3.. Three-dimensional finite element implementation n a three-dimensional analysis the virtual crac extension has to be applied to a single node point on the crac front for evaluating the local value of the energy release rate. For the 0-node three-dimensional isoparametric element, a new crac front is defined by the quadratic interpolation function as shown in Fig. 4. The volume V is identified with the collection of elements which contain the line L c. Thus, in the finite element framewor, for considering the increase in craced area due to the shift of a given particular node M we tae L c to be the line connecting the nodes M, M M + for mid nodes the nodes M -, M -, M, M + M + for corner nodes as shown in Fig. 4. Then, the weight function is therefore taen as follows (Li at al., 985) 0 q = N Q (3) i = where N is the triquadratic shape function Q is the nodal values for the -th node. Note that Q = 0 if the -th node is on S 0. For nodes inside V, Q is given by interpolation between the nodal values on L c S 0 (see Fig. ). 4. Numerical examples discussion The procedures just outlined have been programmed for studying axisymmetric penny shaped cracs three-dimensional cracs under mixed mode loading in isotropic solids. The numerical calculations for mixed-mode crac problems are carried out with the finite element code ABAQUS (Hibbitt et al., 997). This code supplies the required displacements stresses at the Gaussian points to a separate program developed for evaluating the integral the two-state integrals. For the convenience of the notation, we use x, y, z in the numerical experiments instead of x, x x 3 in the previous sections, respectively.

14 4.. An axisymmetric sub-interface penny shaped crac To illustrate the decomposition method for axisymmetric mixed-mode cracs, an axisymmetric sub-interface penny shaped crac is selected as an example problem. The problem of a straight crac paralleling an interface between two dissimilar materials has attracted a substantial amount of attention due to its potential application in various inds of bonding problems (Hutchinson et al., 987; Yang im, 993). n this paper the problem of a penny shaped crac paralleling an interface between two dissimilar materials under tension is investigated. The loading condition the crac geometry under investigation are shown in Fig. 5a ( σ 5 t = N / m ). The cylinder has a radius b = m ; total length of L = 0. 7 m. The penny shaped crac has a radius a = m ( a / b = ); a distance from the interface to the penny shaped crac is h = m. Each material is taen to be isotropic elastic. The upper part of the cylinder has elastic properties of E 9 = N / m ν =0.3. The lower part containing a penny shaped crac has elastic properties of E = N / m ν =0.3. To examine the accuracy the convergence of the results, a typical finite element mesh for a penny shaped crac paralleling an interface between two dissimilar materials is selected as shown in Fig. 5b wherein a total of 98 eight noded isoparametric-axisymmetric elements are used. To find a Mode auxiliary field, we consider a body of the same geometry as in the target problem, but composed of the lower material ( E = E ν = ν ) only. Because the body is made of a homogeneous material, it will undergo a pure Mode deformation under a remote tension σ 5 t = N / m. Note that the foregoing mode decomposition scheme is applicable only when an auxiliary field is found for the target problem, i.e. the geometry the material properties should coincide between the auxiliary field the target problem. Hence we restrict our attention to the lower part of the target field, which has the same geometry properties as the auxiliary field. This is equivalent to removing the upper part applying the external traction, which is acting upon the lower material by the upper one, so that the mode decomposition scheme is now applied on the lower part of the body. t is under a mixed mode in the target problem while it is under pure Mode in the auxiliary field. The first integration domain is comprised of 6 elements adjoining the crac tip; the second 3

15 domain has the next 6 elements adjoining the first integration domain. n this manner, the integration domains 3 through 9 are continued. The detailed element arrangement around the crac tip the domains selected for integration are given in Fig. 6 wherein the paths of the integration domains 3 7 are indicated. The calculations for integrals are carried out according to the domain integral (3) in a separate post-processing program, the two-state integrals are calculated in the same manner. The path independence of two-state integral has been checed numerically by using different domains of integration. The values of two-state integral as calculated by the discrete domain formula of the type of Eq. (3) for the various domains are listed in Table. We observe that the variation in the computed two-state integral from one domain to another is within percent excluding the near tip domain of the first domain. Effects of E / E are examined by taing various material constant E or E / E for the given lower part containing a penny shaped crac ν = ν =0.3. A plot of the variation of the computed with the increase of E / E is shown in Fig. 7. This shows well the mixed mode of the sub-interface penny shaped crac even under uniaxial tension loading, because always higher than are of the same order of magnitude. t is noticed that in the entire range of / E increase as / E E decreases. is E examined. n the case of E / E <, remain relatively unchanged with E / E greater than 30. Note that disappears for E / E =. This is found to be =.089 σ t a, this is well compared with =.903 σ t a for an axisymmetric crac in an infinite homogeneous body from Benthem oiter (973). 4.. Three-dimensional penny shaped crac under nonaxisymmetric loading We first compare the results from the decomposition method for three-dimensional mixedmode cracs with those for the axisymmetric mixed mode crac in the previous section. We consider the full three-dimensional model consisting of a 360 o revolution of the same axisymmetric model as in the previous section about the z axis. The model is discretized with ten elements along the circumference in an 80 o segment. The typical finite element mesh for the full three-dimensional model is similar to that illustrated in Fig. 8. A total of 5740 twenty noded isoparametric elements are used. The elements adjoining the crac front are set to possess the inverse square root singularity at the corner nodes on the crac tip by 4

16 choosing the mid nodes to be located on the quarter point. We briefly discuss the setting up of the integration domains for calculating twostate integrals associated with a unit virtual advance of a finite crac front segment (e.g. see Fig.8). The first integration domain is comprised of a collection of elements adjoining the crac front: the domain consists of two layers of elements in the circumferential direction, amounting to a total of 3 elements, for a unit virtual advance of a corner node, while it contains only one layer of elements or a total of 6 elements if a unit virtual advance is imposed on a mid node. The second domain is obtained by adding one ring of elements around the first domain; thus the second domain has a total of 64 elements for a virtual advance of a corner node a total of 3 elements for a virtual advance of a mid node. n this manner, the integration domains 3 through 9 are continued. The loading condition the crac geometry under investigation are the same as those in 9 the previous part ( E = N / m, E = N / m ν = ν =0.3). As an auxiliary solution, Mode loading condition was used for the cylinder of a homogeneous material E = E = N / m. The results for three-dimensional mixed-mode crac are shown in Table. t is seen from this table that the the two-state integral are pathindependent except the first domain. n Table 3, they are compared with the results from the axisymmetric analysis of the foregoing section. Comparison of the results shows that the three-dimensional solution is in an excellent agreement with the axisymmetric solution. Particularly, the path independence in different annular regions containing crac front at its center for domain integrals indicates the minor effects of boundary. n the three-dimensional solution the the two-state integral for a virtual advance of a corner node differ only slightly from those for a virtual advance of a mid node. To illustrate the mode separation for three-dimensional mixed-mode cracs, selected are two examples of a circular cylinder containing a penny-shaped crac a half-circular cylinder containing a half-penny shaped surface crac at its center under nonaxisymmetric loading. The crac geometry the finite element mesh are shown in Fig 8. The cylinder has a radius b = m ; total length of L = 0. 5 m. The penny shaped crac has a radius a = m ( a / b = 0. 5). For auxiliary solutions, two different loadings are chosen; one is a uniform tension, generating a pure Mode deformation, the other a torsion, giving rise to a pure Mode. To apply nonaxisymmetric loading, the lower end face of the circular 5

17 cylinder at z = L is constrained against motion in the x, y z, whereas the uniform displacement boundary condition of 7 =.54 0 m u z = m are applied u x 6 on the other end face of the circular cylinder at z = L (Fig. 8). The solution procedure starts from the choice of the independent auxiliary fields, which are obtained by the choice of the two different loadings uniform tension torsion. For the auxiliary field of Mode, the same finite element model as that for the target field is used under tension with σ 5 t = N / m, for the auxiliary field of Mode the same finite element model under torsion with the applied torque of 4.6 g m. With both of the auxiliary field data the target field data from ABAQUS output file the evaluation of the two-state integral is carried out for the domain selected as described in this section. A plot of the variation of the computed ( ),, ( ) ( ) ( ) along the crac front of the penny shaped crac is shown in Fig. 9. n the plot, ( ) at the intersection of the crac boundary with the x axis is slightly greater than that at the intersection of the crac boundary with the y axis. is constant along the crac front of the penny shaped crac ( ) as expected. Note that ( ) ( ) vary lie the curves for the squares of cosθ sinθ ( ) along the crac front line of the penny shaped crac. t is noticed that is greatest at the intersection of the crac boundary with the x axis vanishes at the intersection of the ( ) crac boundary with the y axis. On the other h, vanishes at the intersection of the crac boundary with the x axis is greatest at the intersection of the crac boundary with the y axis. This is consistent with the applied displacements of of the cylinder. With these values of the two-state integrals,, u x u z on the top face from Eqs. (), (3) (4). A plot of the variation of the computed are obtained, along the crac front is shown in Fig. 0, in which the absolute values of the stress intensity factors neglecting sign change are plotted. Note that line of the penny shaped crac. As expected, is constant along the crac front vary lie curves of cosθ sinθ respectively along the crac front line of the penny shaped crac. Each of the stress intensity factors, have the variation consistent with that of each of, ( ) ( ) ( ) along the circumferential direction. 6

18 The crac geometry the finite element mesh for the second example are shown in Fig. The half-circular cylinder has a radius b = m ; total length of L = 0. 5 m. The half-penny shaped surface crac has a radius a = m ( a / b = 0. 5), which is of the same size as in the foregoing example. For auxiliary solutions, the same auxiliary field solutions as in the previous example are used. To apply nonaxisymmetric loading, the lower end face of the half-circular cylinder at z = L is constrained against motion in x, y z, whereas the same uniform displacement boundary condition of 7 =.54 0 m u z = m as in the u x 6 previous example are applied on the other end face of the half-circular cylinder at z = L. A plot of the variation of the computed ( ),, ( ) ( ) ( ) along the crac front of the half-penny shaped surface crac is shown in Fig.. n the plot, ( ), at the surface are greater than those at the deepest interior point of the crac front, which is consistent with the numerical experimental results obtained by Ernst et al. (994) Sharobeam Les (995). A plot of the variation of the computed along the crac front is shown in Fig. 3. Note that than those at the deepest interior point of the crac front. As expected,, ( ) ( ) at the surface are greater vary lie the curves of cosθ sinθ, respectively along the crac front line of the half-penny shaped surface crac Mode decomposition using plane-strain asymptotic auxiliary fields Next, we consider the mode decomposition using auxiliary fields of the asymptotic solutions for the plane problems. Nahta Moran (993) Gosz et al. (998) have used the asymptotic auxiliary fields to determine the stress intensity factors for three-dimensional mixed-mode cracs. n their wors, they presented a method to evaluate the divergence term in the two-state integral with the aid of deformation gradients using curvilinear coordinates. n this paper, we proposed a simple method to calculate the two-state integral through imposing displacements of the two-dimensional asymptotic solutions on the nodes in finite element models. As explained before, the divergence term can be easily evaluated with a simple form (9) or (30). The penny shaped crac embedded in a cylinder under mixed mode is adopted to study the mode decomposition using asymptotic auxiliary fields. We tae the same material geometry as those in the previous part ( E = N / m, v = 0. 3, a = m, 7

19 b = m ). The bottom surface is fixed not to move in any direction, the displacements 7 =.54 0 m u z = m are applied on the top surface. As u x 6 a reference solution to verify the effectiveness of the present method, the mode decomposition using numerical auxiliary solutions under pure Mode Mode, which is described in the previous part, is carried out for two types of finite element models - the first model with 7 rings along the crac front as shown in Fig. 8, the second model with 4 rings along the crac front as shown in Fig. 4. The stress intensity factors versus the angle from the x axis, using numerical auxiliary fields for these two finite element models, are plotted in Fig. 5. n this figure, we distinguish the sign change of the stress intensity factors. Since the results for the models with different mesh sizes show a good agreement with each other, the stress intensity factors obtained using numerical auxiliary fields are taen as reference solution to compare with the results obtained using the asymptotic auxiliary fields. Moreover, we have demonstrated the validity of the numerical auxiliary fields in the previous parts. n the comparison, the finite element model with 4 rings is used to examine the mode decomposition employing the asymptotic auxiliary fields in a coarse mesh. Here, the three stress intensity factors,, are obtained from the two-state integrals with three asymptotic auxiliary fields for Mode, Mode, Mode, respectively. The solution is compared with the reference solution obtained by using the numerical auxiliary fields. The stress intensity factors calculated in the areas containing annular rings for domain integral (5) along the crac front show path independence with respect to the integration paths or rings. n Fig. 6, the average values of the stress intensity factors are plotted in terms of the angle from x axis for four solutions: the stress intensities mared by a is the reference solution of Fig. 5 (4 rings), that is, with the numerical solution being taen for the auxiliary field; those indicated by b, c, d are the solutions with plane strain asymptotic field being taen for the auxiliary solution. Furthermore b indicates the solution obtained with the divergence term being neglected, c those obtained by the method of Gosz et al. (988), d by the present method. n this figure, the results obtained by neglecting the divergence term in the two-state integral are a little different from the reference values. One would expect that the divergence term in the two-state integral cannot be ignored for a highly curved threedimensional crac. Note that the present solution is almost indistinguishable from the reference solution while the solution based upon the method of Gosz et. al (998) has some deviations in. Considering the fact that the present method provides a more 8

20 straightforward scheme for calculating the divergence term (9) in the two-state integral, it may be more attractive in view of numerical computation. 5. Concluding remars A method of analysis, based on the conservation laws of elasticity the fundamental relationships in fracture mechanics, has been proposed for studying axisymmetric mixedmode cracs three-dimensional mixed-mode cracs. The method is based on the path independence of two-state integral. Path independence of two-state integral enables us to compute the individual stress intensity factors accurately efficiently from the domain integral expression. The solution procedure has been established shown to be computationally efficient operationally simple: it involves only the choice of appropriate auxiliary solutions in the form of numerical solutions or the plane-strain asymptotic solution, the subsequent calculation of two-state integrals with the aid of the domain integral expression. The asymptotic auxiliary fields for the plane-strain problems are successfully implemented to decompose three-dimensional mixed-mode crac, a simple method is proposed to evaluate the divergence term in the two-state integral. The results for the stress intensity factors decomposed by the present method are found to be in an excellent agreement with the reference solutions obtained with the numerical auxiliary fields. Acnowledgements The authors gratefully acnowledge the financial support by the orea Research Foundation (Grant No E00049) in the course of this study. References Benthem,.P. oiter, W.T., 973, Asymtotic approxiations to crac problems. n: Method of Analysis Solutions of Crac Problems, Noordhoff, Holl. Budiansy, B., Rice,.R., 973, Conservation laws energy release rate. ASME ournal of Applied Mechanics 40, Bui, H.D., 983, Associated path independent integrals for separating mixed modes. ournal 9

21 of the Mechanics Physics of Solids 3, Chen, F.H.., Shield, R.T., 977, Conservation laws in elasticity of the integral type. ournal of Applied Mathematics Physics (ZAMP) 8, -. Choi, N.Y., Earmme, Y.Y., 99, Evaluation of stress intensity factors in circular arcshaped interfacial crac using L integral. Mechanics of Materials, Ernst H.A., Rush P.. McCabe D.E., 994, Resistance curve analysis of surface cracs. n: Fracture Mechanics, Twenty-Fourth Volume. ASTM STP 07, pp Eshelby,.D. (956) The continuum theory of lattice defects. Solid State Physics 3, Gosz, M., Dolbow,. Moran, B., 998, Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracs. nternational ournal of Solids Structures 35, Hartranft, R.. Sih, G.C., 973, Solving edge surface crac problems by an alternating method, method of analysis solutions of crac problems. Noordhoff, Holl. Hibbitt, H.D., arlsson, B. Sorensen, E.P., 997, ABAQUS user s manual. Hibbitt, arlsson Sorensen, Providence, R. Hong, C.C., Stern, M., 978, The computation of stress intensity factors in dissimilar materials. ournal of Elasticity 8, -30. Hutchinson,.W., 968, Singular behavior at the end of a tensile crac in a hardening material. ournal of the Mechanics Physics of Solids 6, 3-3. Hutchinson,.W., Mear, M.E., Rice,.R., 987, Crac paralleling an interface between dissimilar materials. ASME ournal of Applied Mechanics 54, m, S., im,.s., 000, Application of the two-state M integral for computing an intensity factor of a singular near-tip field for a generic composite wedge. ournal of the Mechanics Physics of Solids 48, 9-5. nowles,.., Sternberg, E., 97, On a class of conservation laws in linearized finite elastostatics. Archives of Rational Mechanics Analysis 7, assir, M.. Sih, G.C., 974, Three dimensional crac problems. n: Mechanics of Fracture, Vol.. Noordhoff, Holl. uo, A., 987, On the use of a path-independent line integral for axisymmetric cracs with nonaxisymmetric loading. ASME ournal of Applied Mechanics 54, Li, F.Z., Shih, C.F., Needleman, A., 985, A comparison of methods for calculating energy release rates. Engineering Fracture Mechanics,

22 Moran, B., Shih, C.F., 987, Crac tip associated domain integrals from momentum energy balance. Engineering Fracture Mechanics 7, Nahta, R. Moran, B., 993, Domain integrals for axisymmetric interface crac problems. nternational ournal of Solids Structures 30, Naamura, T. Pars, D.M., 989, Antisymmetrical 3-d stress field near the crac front of a thin elastic plate. nternational ournal of Solids Structures 5, Naamura, T., 99, Three-dimensional stress fields of elastic interface cracs. ASME ournal of Applied Mechanics 58, Niishov, G.P. Atluri, S.N., 987a, Calculation of fracture mechanics parameters for an arbitrary three dimensional crac by the equivalent domain integral method. nternational ournal for Numerical Methods in Engineering 4, Niishov, G.P. Atluli, S.N., 987b, An equivalent domain integral method for computing crac tip integral parameters in nonelastic, thermo-mechanical fracture. Engineering Fracture Mechanics 6, Rice,.R., 968, A path-independent integral approximate analysis of strain concentrations by notches cracs. ASME ournal of Applied Mechanics 35, Sharobeam M.H. Les.D., 995, A single specimen approach for -integral evaluation for semi-elliptical surface cracs. n: Fracture Mechanics, Twenty-Fifth Volume. ASTM STP 0, pp Shih, C.F., Asaro, R.., 988, Elastic-plastic analysis of cracs on bimaterial interfaces: Part -structure of small-scale yielding fields. ASME ournal of Applied Mechanics 55, Stern, M., Becer, E.B., Dunham, R.S., 976, A contour integral computation of mixed mode stress intensity factors. nternational ournal of Fracture, Wang, S.S., Yau,.F., 98, nterfacial cracs in adhesively bonded scarf joints. AAA 9, Yang, M., im,., 993, The behavior of sub-interface cracs with crac-face contact. Engineering Fracture Mechanics 44, Yau,.F., Wang, S.S., Corten, H.T., 980, A mixed mode crac analysis of isotropic solids using conservation laws of elasticity. ASME ournal of Applied Mechanics 47,

23 Table Axisymmetric analysis result for a sub-interface penny shaped crac

24 Table Three-dimensional analysis results for a sub-interface penny shaped crac 3

25 Table 3 Comparison between the axisymmetric analysis result the three-dimensional analysis results 4

26 Fig.. Simply connected region A enclosed by the contour C on the cross section of an axisymmetric crac. 5

27 (a) (b) (c) Fig.. (a) Conventions at curvilinear crac front (b) virtual crac advance between s b (c) inner tubular surface S t outer arbitrary surface S 0. s a 6

28 Fig. 3. Local cylindrical coordinates on the crac front; p is the projection point of p on the plane parallel to the crac surface. 7

29 Fig. 4. Local advance of the crac front for 0-noded three-dimensional element. 8

30 (a) (b) Fig. 5. (a) Axisymmetric sub-interface penny shaped crac (b) finite element model for axisymmetric sub-interface penny shaped crac. 9

31 Fig. 6. The detailed element arrangement around the crac tip the paths of the integration domains. 30

32 Fig. 7. The variation of with the increase of / E E ( ν = ν =0.3). 3

33 Fig. 8. (a) Three-dimensional finite element model for the penny shaped crac under nonaxsymmetric loading (b) the definition of q along the penny shaped crac. i 3

34 Fig. 9. The calculated pointwise values of ( ), ( ), ( ) of the penny shaped crac. along the crac front ( ) Fig. 0. The variation of the computed, penny shaped crac. along the crac front of the 33

35 Fig.. Three-dimensional finite element model for half-penny shaped surface crac under nonaxsymmetric loading. 34

36 ( ) ( ) ( ) ( ) Fig.. The calculated pointwise values of,, along the crac front of the half-penny shaped surface crac. Fig. 3. The variation of the computed, along the crac front of the halfpenny shaped surface crac. 35

37 Fig. 4. Three-dimensional finite element model with 4 rings along the crac front. 36

38 Calculated value (Pa-m / ) ring 7ring 4ring 7ring 4ring 7ring Counterclocwise angle from the x axis (degree) Fig. 5. Comparison of stress intensity factors calculated by using numerical auxiliary fields for two types of meshes with 4 rings with 7 rings. 37

39 a a a b b b c c c d d d 00 Calculated value (Pa-m / ) Counterclocwise angle from the x axis (degree) Fig. 6. Mode decomposition of the penny shaped crac under mixed mode in Fig. 4. The stress intensity factor mared by superscript a is the solution of Fig. 5 when numerical solution is used for the auxiliary field. Those indicated by b, c, d are the solutions when the plane strain asymptotic solutions are employed for the auxiliary field: b : the divergence term being neglected in equation (5); c : obtained by the method from Gosz et. al (998); d : the present method. 38